imprint plane with boolean modifier
I see 2 solutions for your problem:
If you want to apply your modifiers, you can extrude your plane to a big cube (bigger than the cube you want to imprint). Apply the boolean modifier, and delete the extruded faces.

If you want to keep your modifiers, you will have to create some thickness.
Start with a slightly bigger cube you will add (modifier -> boolean -> union operation), then difference boolean with a  modified cube (with larger part over your plane to cut the upper part of the initial cube, and dig in your plane).